Products in spin$^c$-cobordism

We calculate the mod $2$ spin$^c$-cobordism ring up to uniform $F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $\leq 33$. We construct an infinitely generated nonunital subring of the $2$-torsion in the spin$^c$-cobordism ring. We use our calculations of product structure in the spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism, of a compact $24$-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor.